How do I bend a plane and keep the objects on it following its bend?
In edit mode subdivide the plane so to have a grid with enough vertices that will define the position of your cube, then select the 3 vertices nearest the position you want the cube to stay.

Then in object mode select the cube, Shift
 select the plane, press Ctrl+P
 and choose "vertex triangle", so the cube will always follow the average location of the three vertices of the plane.